# Heat transfer in the thin-film region of an axially-grooved structure

Figure 1 Thin evaporating film on a fragment of the rough solid surface (Khrustalev and Faghri, 1995; A and B corresponding to points shown on Fig. 2).
Figure 2 Cross-section of the characteristic element of (a) an axially-grooved condenser, and (b) an evaporator (Khrustalev and Faghri, 1995; A and B are shown in Fig. 1).
Figure 2 Cross-section of the characteristic element of (a) an axially-grooved condenser, and (b) an evaporator (Khrustalev and Faghri, 1995; A and B are shown in Fig. 1).

Khrustalev and Faghri (1995) modeled an evaporating extended meniscus in a capillary groove, shown in Fig. 2(b). Most of the heat is transferred through the region where the liquid layer is extremely thin. The significance of the temperature difference between the saturated vapor core and the interface has been stressed in the model. Manufacturing processes always leave some degree of roughness on metallic surfaces. Alloys of copper, brass, steel, and aluminum invariably have some distinct grain structure resulting from processing. Corrosion and deposition of some substances on the surface can further influence its microrelief. This means that the solid surface of a working material is totally covered with microroughnesses, where the characteristic size Rr may vary from 10-8 to 10-6 m, for example. Apparently, thin liquid film formation can be affected by some of these microroughnesses. It can be assumed that at least some part of a single roughness fragment on which thin film formation takes place has a circular cross-section and is extended in the z-direction due to manufacturing of the axial grooves (see Fig. 1).

As can be seen in Fig. 1, the free liquid surface is divided into four regions. The first region is the equilibrium nonevaporating film. The second (microfilm) region ranges in the interval ${{\delta }_{0}}\le \delta \le {{\delta }_{1}}$, where the liquid film thickness increases up to the value δ1 as described by eqs. (15) and (16). In this region, the generalized capillary pressure

 ${{p}_{\text{cap}}}\equiv \sigma K-{{p}_{d}}$ (1)

(defined so that its value is positive) changes drastically along the s-coordinate. The capillary pressure changes from the initial value up to an almost-constant value at point s1, where the film thickness δ1 is large enough to warrant neglecting the capillary pressure gradient. It is useful to mention that some investigators have denoted this microfilm region as the interline region. The third (transition) region, where the liquid-vapor interface curvature is constant, is bounded by δ1 < δ < Rr + δ0, and the local film thickness is determined by the geometry of the solid surface relief and the value of the meniscus radius Rmen. In the fourth (meniscus) region, where δ > Rr + δ0 by definition, the local film thickness can be considered independent of the solid surface microrelief. In the third and fourth regions, heat transfer is determined by heat conduction in the meniscus liquid film and the metallic fin between the grooves. In the second region, however, the temperature gradient in the solid body can be neglected in comparison to that in the liquid due to the extremely small size of this region.

The total heat flow rate per unit groove length in the microfilm region is defined as

 ${{{q}'}_{mic}}\left( {{s}_{1}} \right)=\int_{0}^{{{s}_{1}}}{\frac{{{T}_{w}}-{{T}_{\delta }}}{\delta /{{k}_{\ell }}}ds}\equiv \int_{0}^{{{s}_{1}}}{{q}''ds}$ (2)

Equations (15) – (17), (18) and (19) must be solved for four variables: δ, dδ / ds, pcap and q'mic(s) in the interval from s = 0 to the point s = s1, where pcap can be considered to be constant. Instead of considering the two second-order equations (15) and (16), the following four first-order equations should be considered with their respective boundary conditions:

 $\frac{d\delta }{ds}=\delta '$ (3)

 $\frac{d\delta '}{ds}={{\left( 1+\delta {{'}^{2}} \right)}^{3/2}}\left( \frac{{{p}_{\text{cap}}}-A'{{\delta }^{-B}}}{\sigma }+\frac{1}{{{R}_{r}}} \right)$ (4)

 $\frac{d{{p}_{cap}}}{ds}=-\frac{3{{\nu }_{\ell }}}{{{h}_{\ell v}}{{\delta }^{3}}}{{{q}'}_{mic}}\left( s \right)$ (5)

 $\frac{d{{{{q}'}}_{mic}}}{ds}=\frac{{{T}_{w}}-{{T}_{\delta }}}{\delta /{{k}_{\ell }}}$ (6)

 ${{\left. \delta \right|}_{s=0}}={{\delta }_{0}}$ (7)

 ${{\left. \delta ' \right|}_{s=0}}=0$ (8)

 ${{\left. {{p}_{\text{cap}}} \right|}_{s=0}}=-\frac{\sigma }{{{R}_{r}}+{{\delta }_{0}}}+A'\delta _{0}^{-B}$ (9)

 ${{\left. {{{{q}'}}_{mic}} \right|}_{s=0}}=0$ (10)

where the value of δ0 can be found from eq. (20) with K = − 1 / Rr.

Though the initial-value problem described by eqs. (3) – (10) is completely determined, its solution must satisfy one more condition – namely,

 ${{\left. {{p}_{\text{cap}}} \right|}_{s={{s}_{1}}}}=\frac{\sigma }{{{R}_{\text{men}}}}$ (11)

Since the only parameter in this problem that is not fixed is connected with the surface roughness characteristics, the boundary condition (11) can be satisfied by the choice of Rr. Physically, this means that the beginning of the evaporating film shifts along the rough surface depending on the situation so as to satisfy the conservation laws. However, in a smooth surface model $\left( {{R}_{r}}\to \infty \right)$ the solution will probably not satisfy eq. (11). As a result of this problem, the values of δ1 and q'mic(s1)can be determined and the transition region can be considered provided that δ1 < Rr, where the free liquid surface curvature is constant and its radius Rmen is many times larger than Rr. Based on the geometry shown in Fig. 2,

 $\delta ={{\delta }_{0}}+{{R}_{r}}-\sqrt{R_{r}^{2}-{{x}^{2}}}-{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{x}^{2}}+2{{R}_{\text{men}}}x\sin {{\theta }_{f}} \right)}^{1/2}}$ (12)

where θf is the contact angle in the microfilm region. Equation (12) is valid for the rough surface model ( θf can be set equal to zero for very small Rr) as well as the smooth surface model in the meniscus region (${{R}_{r}}\to \infty$ and θf is given as a result of the microfilm problem solution).

The heat flow rate per unit groove length in the transition region is

 ${{{q}'}_{tr}}=\int_{{{x}_{f}}}^{{{x}_{tr}}}{\frac{{{T}_{w}}-{{T}_{\delta }}}{\delta /{{k}_{\ell }}}dx}$ (13)

where xf and xtr can be obtained from eq. (12), provided δ = δ1 and δ = Rr + δ0, respectively.

At this point, the point of connection between the transition and meniscus regions must be considered. Here the film thickness, free surface curvature, and liquid surface slope angle must coincide on both sides. In the rough surface model, the last condition is always satisfied because the length of the microfilm region is smaller than Rr and the rough fragment with the film can be “turned” around its center in the needed direction. In other words, because of the circular geometry of the rough fragment and the constant temperature of the solid surface in the microfilm region, the slope of the film free surface is not fixed in the mathematical model. On the contrary, in the smooth surface model the numerical results give θf, which is generally not equal to θmen as determined by the fluid flow along the groove. Note that in a situation where ${{\theta }_{f}}\ne {{\theta }_{\text{men}}}$, the smooth surface model can be used along with the rounded fin corner, where the radius is Rfin. In this type of case eq. (12) can also be used, provided Rr is changed to Rfin.

It is useful to note that the values of Rmen and θmen are connected by the geometric relation ${{\theta }_{\text{men}}}=\arccos \left( W/2{{R}_{\text{men}}} \right)-\gamma$. These values should be given as a result of the solution of the problem for fluid transport along the groove. The fin top temperature Tw should be defined based on consideration of the heat conduction problem in the fin between the grooves and in the meniscus liquid film, as discussed below.

The free liquid surface curvature K in the microfilm region varies from the initial value to that in the meniscus region. Its variation is described by eqs. (3) – (11) with respect to the pcap and pd definitions. In spite of a sharp maximum, which the K function has in the microfilm region, its variation only slightly affects the total heat transfer coefficient. To check this hypothesis numerically, a simplified version of the heat transfer model of the microfilm region was developed by Khrustalev and Faghri (1995), wherein it was assumed that the microfilm free surface curvature is equal to that in the meniscus region. Therefore, instead of solving eqs. (3) – (11), the microfilm thickness in this region (and in the transition region) can be given by eq. (12) for the interval $0\le x\le {{x}_{tr}}$. In this case, the heat flow rate per unit groove length in both the microfilm and transition regions is

 ${{{q}'}_{\text{mic}}}+{{{q}'}_{tr}}=\int_{0}^{{{x}_{tr}}}{\frac{{{T}_{w}}-{{T}_{\delta }}}{\delta /{{k}_{\ell }}}dx}$ (14)

Khrustalev and Faghri (1995) solved eqs. (18) and (19) simultaneously for Tδ and ${{\left( {{p}_{\text{sat}}} \right)}_{\delta }}$

for every point on s. The system of the four first-order ordinary differential equations with four initial conditions and one constitutive condition describing the evaporating microfilm region, eqs. (3) – (11), were solved using the fourth-order Runge-Kutta method and the shooting method (on parameter Rr). The controlled relative error was less than 0.001 for each of the variables. The results obtained for comparatively small temperature drops through the thin film were compared with those from the simplified model.

Characteristics of the evaporating film along the solid-liquid interface (ammonia, Tv=250 K): (a) Free liquid surface temperature; (b) Thickness of film; (c) Generalized capillary pressure (Khrustalev and Faghri, 1995).

The results presented in this section mostly refer to the AGHP with a total length of Lt = 0.914 m, a condenser length of Lc = 0.152 m, and an evaporator length of $0.15\le {{L}_{e}}\le 0.343$ m; other geometric parameters are: W = 0.61 mm, tg = 1.02 mm, L1 = 0.43 mm, Rv = 4.43 mm, Ro = 7.95 mm, γ = 0o, and N = 27. The working fluids were ammonia and ethane, and the thermal conductivity of the casing material was assumed to be kw = 170 W/m-K, with dispersion constants A' = 10 − 21J and B = 3.

The data in Figs. 3 and 4 were obtained for ammonia in the evaporator with a vapor temperature of Tv = 250 K and α = 1. The solid surface superheat was $\Delta T=\left| {{T}_{w}}-{{T}_{v}} \right|$, and results obtained using the simplified model for evaporating film are denoted as SIMPL. Figure 3(a) shows the variation of the free liquid surface temperature along the evaporating film for ΔT = 0.047 K, 0.070 K, and 0.120 K, which values are from the solutions of eqs. (21), (18), (19) and (3) – (11) in the microfilm region. These results are compared to those obtained by the simplified model, which is based upon the assumption that the curvature of the microfilm free surface is equal to that of the meniscus region. In the simplified model for the case of a smooth surface, the value of the contact angle in the microfilm region was θf = 7o; this value was obtained by the numerical solution of eqs. (3) – (11).

The corresponding variations in the film thickness δ and generalized capillary pressure pcap are shown in Figs. 3 (b) and (c). The results obtained by the simplified model have been artificially shifted along the s-coordinate in these figures (and also in Fig. 1) to make the comparison more understandable. Also, it should be noted that there is some difference between the s-coordinate and the x-coordinate used in the simplified model. The following relation has been used in the present analysis: s = Rrarcsin(x / Rr).

Figure 4 Heat flux through the evaporating film (ammonia, Tv=250 K, α =1): (a) along the solid-liquid interface (microfilm region); (b) along the fin axis (Rr=33 μm, ΔT = 1 K) (Khrustalev and Faghri, 1995)

In Fig. 3(a), the interval of Tδ variation along the evaporating film from the value of Tw¬ to approximately Tv is very prolonged, and the interfacial thermal resistance is significant even when the film thickness is larger than 0.1 μm. For a smaller characteristic size R, the film thickness increased more sharply along the solid surface, as shown in Fig. 3(b), which is in agreement with eq. (12). It should be mentioned that unlike for the simplified model, the problem defined by eqs. (3) – (11), Rr is not a parameter but the result of the numerical solution. The values of the maximum heat flux in the microfilm region were extremely high in comparison to those in the meniscus region (see Fig. 4). For ΔT = 0.120 K, the generalized capillary pressure pcap decreased from the initial value to an almost-constant value by approximately 5000 times [see Fig. 3(c)]. For a larger ΔT, this sharp decrease can cause some difficulties in the numerical treatment while solving eqs. (3) – (11); that is why the simplified model is useful. The simplified model has given the variation of pcap along the film, which is even more drastic because the surface tension term is absent in the capillary pressure gradient [see Fig. 3(c)]. However, this assumption caused a comparatively small decrease in total heat flow rate in the microfilm region, as illustrated by Fig. 2(a). The distribution of the heat flux in the microfilm, transition and beginning of meniscus regions for different meniscus contact angles θmen – as predicted by the simplified model – are presented in Fig. 4(b). The total heat flow through the meniscus region was significantly larger than that through the microfilm region. This means that the simplified model should provide the needed accuracy when estimating the heat transfer coefficient for an evaporator element, as shown in Fig. 2. Khrustalev and Faghri (1995) showed that the simplified model underestimated the overall heat transfer coefficient by only 5%, which permits its use as a means of avoiding the numerical difficulties described above.

## Contents

 $\frac{1}{3{{\mu }_{\ell }}}\frac{d}{ds}\left[ {{\delta }^{3}}\frac{d}{ds}\left( {{p}_{d}}-\sigma K \right) \right]=\frac{{{k}_{\ell }}\left( {{T}_{w}}-{{T}_{\delta }} \right)}{{{h}_{\ell v}}{{\rho }_{\ell }}\delta }$ (15)
 $K={{K}_{w}}+\frac{{{d}^{2}}\delta }{d{{s}^{2}}}{{\left[ 1+{{\left( \frac{d\delta }{ds} \right)}^{2}} \right]}^{-3/2}}$ (16)
 pv = psat(Tv) (17)
 ${{({{p}_{sat}})}_{\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left[ \frac{{{({{p}_{sat}})}_{\delta }}-{{p}_{sat}}({{T}_{\delta }})+{{p}_{d}}-\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right]$ (18)
 ${{T}_{\delta }}={{T}_{w}}+\frac{\delta }{{{k}_{\ell }}}\left( \frac{2\alpha }{2-\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v}}}{\sqrt{{{T}_{v}}}}-\frac{{{({{p}_{sat}})}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]$ (19)
 ${{p}_{v,\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left\{ \frac{[{{p}_{v,\delta }}-{{p}_{sat}}({{T}_{\delta }})-{{p}_{cap}}+{{p}_{d}}]}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right\}$ (20)
 pd = − A'δ − B (21)

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Khrustalev, D. K., and Faghri, A., 1995, “Heat Transfer during Evaporation and Condensation on Capillary-Grooved Structures of Heat Pipes,” ASME Journal of Heat Transfer Vol. 117, pp. 740-747.